Distribution data of phenomena has been used in various industrial fields. Examples of distributions of phenomena include distributions of physical characteristics and chemical characteristics of materials in the material industry, distributions of customer preferences in product categories, distributions of product prices, and distributions of predictive probability of a certain result in research on needs trends.
For example, data on the distribution of ions implanted into a semiconductor substrate in the substrate depth direction in a semiconductor manufacturing process (hereinafter referred to as ion implantation distribution) may be obtained by experiment or simulation. In general, ion implantation distributions are acquired as a result of implantation under certain conditions. To predict an ion implantation distribution for given conditions different from conditions under which an ion implantation distribution has been obtained, the following procedure is followed.
(1) A mathematical expression that enables analysis of ion implantation distribution data is provided. The mathematical expression represents a model function that agrees with ion implantation distribution data within some margin of error. Here, the model function may also be called approximation function. The model function is a function on coordinate axes where the horizontal axis represents positions in the semiconductor substrate in the depth direction and the vertical axis represents the concentrations of implanted ions. The model function has multiple parameters. Data on ion implantation distribution in at least a certain depth range under different conditions may be estimated by changing the values of the parameters of the model function.
(2) A computer that manages ion implantation distributions stores in a database the relationships between multiple implantation conditions and parameters of the approximation functions that approximate ion implantation distributions under multiple conditions for ion implantation distribution data acquired under the multiple conditions.
(3) The computer determines interpolated values of the parameters for conditions different from the conditions stored in the database on the basis of the relationships between the conditions and the parameters in the database and predicts ion implantation distributions from the approximation function.
Pearson distributions or the dual Pearson distributions have been used as functions for modeling ion implantation distributions. For example, the concentration N(x) of ions of a given material implanted to a depth x in a substrate of a given semiconductor at a given acceleration energy may be expressed by the following dual Pearson distribution:N(x)=(Φ−Φchan)*ha(x)+Φchan*hc(x)  [Expression 1]where Φ is the total dose of the implanted ions, Φchan is the dose of channeling ions. Ion implantation distribution curves often have a peak. A concentration distribution called “tail” which approaches asymptotically to a concentration of 0 is often formed in the semiconductor substrate in the depth direction from the peak. The tail is thought to be caused by implanted ions that snake through gaps between crystal axes in the semiconductor substrate. The tail is therefore also called channeling part and the dose in the tail is also called channeling dose.
Of the total dose Φ, Φ−Φchan is called the dose component in the amorphous part. The amorphous part corresponds to the ion implantation distribution of ions implanted in an amorphous semiconductor substrate. It is thought that in a highly dosed, high-ion-concentration region, crystals in the semiconductor substrate may be destroyed and implanted ions exhibit a distribution similar to that of ions implanted in an amorphous semiconductor substrate. Therefore, the peak region of an ion implantation distribution is called amorphous part.
In Equation 1, ha(x) is a function representing the Pearson distribution corresponding to the amorphous part and hc(x) is a function representing the Pearson distribution corresponding to the channeling part. Equation 1 uses both the Pearson distribution corresponding to the amorphous part and the Pearson distribution corresponding to the channeling part to represent an ion implantation distribution and hence called dual Pearson distribution.
The Pearson distribution is expressed by the function given below. Here, a Pearson IV function among the functions called Pearson function family will be illustrated with an independent variable denoted by x. The Pearson IV function is also called Pearson IV distribution. The Pearson IV function will be hereinafter sometimes simply referred to as Pearson function. The Pearson IV distribution will be sometimes simply referred to as Pearson distribution.
The Pearson function takes various forms depending on the value of (γ2, β). Pearson IV is one of the various forms. While an equation of only Pearson IV will be given here, the same discussion applies to other Pearson functions as well. A Pearson distribution for (γ2, β) is used herein.
                    ⁢          [              Expression        ⁢                                  ⁢        2            ]                  Hma      ⁡              (        x        )              =          K      ⁢                                            b            0                    +                                    b              1                        ⁢            x                    +                                                                                                           b                    2                                    ⁢                                      x                    2                                                                                                                1                                      2                    ⁢                                          b                      2                                                                      ⁢                                                                                        ⁢                          exp              ⁡                              [                                                      -                                          (                                                                                                    b                            1                                                                                2                            ⁢                                                          b                              2                                                                                                      +                        a                                            )                                                        ⁢                                      2                                                                                            4                          ⁢                                                      b                            2                                                    ⁢                                                      b                            0                                                                          -                                                  b                          1                          2                                                                                                      ⁢                                                            tan                                              -                        1                                                              ⁡                                          (                                                                                                    2                            ⁢                                                          b                              2                                                        ⁢                            x                                                    +                                                      b                            1                                                                                                                                                              4                              ⁢                                                              b                                2                                                            ⁢                                                              b                                0                                                                                      -                                                          b                              1                              2                                                                                                                          )                                                                      ]                                                        
Here,
Rp is the parameter representing the projected range of ions in a distribution representing ion implantation in the Pearson IV distribution or a Gaussian distribution. ΔRp is the parameter representing the standard deviation around Rp in the Pearson IV distribution function or Gaussian distribution function, that is, the spread of the distribution;γ is the parameter representing the left-right asymmetry of the ion implantation distribution in Pearson IV;β is the parameter representing the sharpness of the peak of the ion implantation distribution in Pearson IV;Hma(x) is the distribution function of the amorphous part of the ion implantation distribution. The amorphous part is also called main part.x is the depth from the surface of the substrate;K is a normalization constant;A is 10β-12γ2-18;a0 is equal to −ΔRpγ(β+3)/A:b0 is equal to −ΔRp2 (4β−3γ2)/A;b1 is equal to a0;b2 is equal to −(2β−3γ2−6); andD is equal to 4b2bo−b12. Among these parameters, parameters Rp, ΔRp, γ and β are called moment parameters. When the values of the moment parameters (Rp, ΔRp, γ and β) are determined, the value of the Pearson IV distribution function is determined.
FIG. 1 illustrates an exemplary ion implantation distribution in an amorphous silicon substrate. Characteristics of the Pearson distribution itself will be discussed below through discussion of distributions in amorphous crystals, with influences of the channeling part being reduced. In FIG. 1, the horizontal axis represents the depth in the substrate and the vertical axis represents ion concentration measured using a Secondary Ion-microprobe Mass Spectrometry (SIMS). FIG. 1 illustrates results of ion implantations at varying ion (boron) acceleration energies ranging from 20 keV to 80 keV.
As illustrated in FIG. 1, the peak moves in the substrate depth direction as the acceleration energy increases. The rising section of the distribution profile increases with respect to the substrate depth x as the acceleration energy increases. The profile of the rising section of the 80-keV distribution is almost exponential. On the other hand, the ion concentration rapidly decreases in the section deeper than the peak. Accordingly, the distribution profile becomes more asymmetrical about the peak with increasing acceleration energy. The Pearson distribution may also accurately represent a distribution that is asymmetrical and skewed from a Gaussian distribution, like the 80-keV distribution in FIG. 1.
FIG. 2 illustrates dependence of Rp and ΔRp on acceleration energy. FIG. 3 illustrates dependence of γ and β on acceleration energy. FIGS. 2 and 3 illustrate relationships between acceleration energies and the moment parameters (Rp, ΔRp, γ and β), where the sets of moment parameters (Rp, γRp, γ and β) Pearson are determined so that the moment parameters match ion implantation distributions obtained by varying acceleration energy.
As illustrated in FIG. 2, Rp and ΔRp depend on the energy almost linearly. However, the dependence of ΔRp on the energy is small as compared with Rp. In FIG. 3, γ is generally −1.5, which agrees with the fact that the peak of the ion implantation distribution in FIG. 1 is skewed to the deeper side from the center of the distribution range of implanted ions (in the direction opposite to the surface of the substrate). Here, β is approximately 9. FIG. 4 lists exemplary parameter values.
As illustrated in FIG. 1, the Pearson distribution seemingly may represent almost every ion implantation distribution without problems. However, the peaks of the ion implantation distributions in FIG. 1 are skewed to the deeper side from the center of the distribution range of implanted ions. This qualitatively corresponds to the fact that γ is negative. A closer look shows that the ion implantation distribution at an acceleration energy of 20 keV is relatively symmetric, but the skew of the peak of the ion implantation distribution increases as acceleration energy increases. That is, FIG. 1 qualitatively represents that the asymmetry of a distribution increases with increasing acceleration energy. It may be presumed that this may be represented by a negative value of the moment parameter γ which represents asymmetry and by the absolute value of γ that increases with increasing acceleration energy.
However, FIGS. 3 and 4 do not represent such trend. Specifically, the value of γ is generally constant at −1.5. The reason why the value of γ does not necessarily vary as a function of acceleration energy will be described below.
FIG. 5 illustrates an example in which a Pearson distribution is fitted to a 60-keV Boron (B) distribution. It may be seen from FIG. 5 that distributions for different sets of (γ, β) may be represented with nearly equal accuracy. That is, there may exist different sets of (γ, β) for one Pearson distribution. In other words, the combinations of (γ, β) for one Pearson distribution lack uniqueness.
The lack of uniqueness will be a hindrance to building of a database. In order to reflect the trend of the distribution profile as has been described above, γ needs to be decreased in the negative direction (the absolute value of γ needs to be increased) as energy increases. However, there is substantial arbitrariness in determination of the value of γ. For example, the moment parameter may be extracted with a minimal change of β. For γ in a certain range of values, the Pearson distribution accurately represents one distribution by adjusting the value of β according to the value of γ. That is, parameters γ and β interact with each other to a great degree. As energy increases, parameter γ monotonically decreases in the negative direction (the absolute value of γ increases). However, it is difficult to quantitatively determine γ while ensuring the uniqueness of γ in the process of matching the Pearson distribution to ion implantation distribution data.
FIG. 6 illustrates an allowable region (β>β3) for the Pearson function family. Roman numerals I to VI in FIG. 6 denote functions included in the Pearson function family. Curves β3, βb2, and βD2 correspond to conditional expressions that characterize the regions where the functions in the Pearson function family exist. For example, the Pearson IV function is in the range βD2<β.
Combinations (γ, β) in FIG. 3 which are effective for 60-keV boron 3 is represented by the dashed line in FIG. 6. It may be seen from FIG. 6 that a considerably wide range of combinations of values indicated by the dashed line in FIG. 6 are possible under the single condition of 60 keV.    [Patent Document] Japanese Laid-Open Patent Publication No. 2008-124075[Non-patent Document]    A. F. Tasch, H. Shin, C. Park, J. Alvis and S. Novak, “An improved approach to accurately model shallow B and BF2 implants in silicon”, J. Electrochem. Soc. (U.S.A.), 1989, Vol. 136, pp. 810-814    C. Park, K. M. Klein and A. F. Tasch, “Efficient modeling parameter extraction for dual Pearson approach to simulation of implanted impurity profiles in silicon”, Solid-State Electronics, (U.S.A.), 1990, Vol. 33, pp. 645-650    K. Suzuki, Ritsuo Sudo and T. Feudel, “Simple analytical expression for dose dependent ion-implanted Sb profiles using a joined half Gaussian function and one with exponential tail”, Solid-State Electronics, (U.S.A), 1998, Vol. 42, pp. 463-465    K. Suzuki, R. Sudo, Y. Tada, M. Tomotani, T. Feudel, and W. Fichtner, “Comprehensive analytical expression for dose dependent ion-implanted impurity concentration profiles”, Solid-State Electronics, (U.S.A.), 1998, Vol. 42, pp. 1671-1678    K. Suzuki, R. Sudo, T. Feudel, and W. Fichtner, “Compact and comprehensive database for ion-implanted As profile, “IEEE Trans. Electron Devices, (U.S.A.), 2000, ED-47, No. 1, pp. 44-49    K. Suzuki and R. Sudo, “Analytical expression for ion-implanted impurity concentration profiles”, Solid-State Electronics, Vol. 44, pp. 2253-2257, 2001    W. K. Hofker, “Implantation of boron in silicon”, Philips Res. Rep. Suppl., (Netherlands), 1975, Vol. 8, pp. 1-121    D. G. Ashworth, R. Oven, and B. Mundin, “Representation of ion implantation profiles by Pearson frequency distribution curves”, J. Phys. D., (U.S.A.), 1990, Vol. 23, pp. 870-876    J. F. Gibbons, S. Mylroie, “Estimation of impurity profiles in ion-implanted amorphous targets using joined half-Gaussian distributions”, Appl. Phys. Lett., (U.S.A.), 1973, Vol. 22, p. 568-569
As has been described above, it is difficult to obtain a single unique set of (γ, β) when a Pearson distribution is fitted to phenomenon distribution data. For example, when fitting a Pearson function to distribution data of one phenomenon, arbitrariness of set (γ, β) makes it difficult to store the relationships between conditions under which phenomenon distribution data have been acquired and (γ, β) of the Pearson distribution that matches the distribution data of the phenomenon. It is also difficult to interpolate the values of parameters under conditions other than conditions under which distribution data of a phenomenon has been acquired, on the basis of the relationships between the conditions and the parameters, to predict distribution data for the phenomenon from a Pearson function using interpolated values.